Do you have any of the children's birth certificates to give mother's maiden name?
There is this marriage entry on familysearch beta
31/12/1850 Thornley (near Tow-Law) Durham
John Atkinson age 22 father Henry
Elizabeth Hey age 18 father Thomas
edited:-
probably not your John & Elizabeth
These details for
Elizabeth ILEY christening: 11 Oct 1835 Newlands, Whittonstall, Northumberland
death: 17 Jun 1894 Greencroft Cotts, Annfield Plain, Durham, England
burial: 19 Jun 1894 Harelaw, Annfield Plain, Durham, England
Father Thomas mother Mary Leighton
John Atkinson born 1828 Wingate Grange Kelloe Durham
died 31/1/1892 Pontop Annfield Plain Durham
burial 2/2/1892 Harelaw annfield Plain Durham
parents Henry Atkinson mother Isabella Bullerwell